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%I #24 Jan 02 2023 09:00:32
%S 1,3,154,8369,711226,90349957,16012077362,3768789527617,
%T 1136241039871954,426747190631335301,195301450278484563322,
%U 106968871128338892427537,69076413764424335543681642,51931946172675368683512111589,44964793280161619728525791864226,44419470206051792513510236597094657
%N Number of equivalence classes of pairs of permutations on S2n where 2 pairs are equivalent if they generate similar maps on Dyck paths.
%H Kevin Limanta, Hopein Christofen Tang, and Yozef Tjandra, <a href="https://arxiv.org/abs/2105.14439">Permutation-generated maps between Dyck paths</a>, arXiv:2105.14439 [math.CO], 2021.
%F a(n) = 1 - n^2 + 2*Sum_{a=1, n-1} Sum_{b=1, n-1} n!^2*(binomial(2*n-2-a-b, n-2)+binomial(2*n-2-a-b, n-1-a))/(max(a,2)!*max(b,2)!) for n>=3. - clarified by _Kevin Limanta_, Dec 29 2022
%F a(n) ~ c * 2^(2*n + 1) * n^(2*n + 1/2) / exp(2*n), where c = sqrt(Pi) * (25/16 + exp(1) - 5*exp(1/2)/2) = 0.281782323432896188420860093697452373839427854773... - _Vaclav Kotesovec_, Dec 29 2022
%t Join[{1,3}, Table[1 - n^2 + 2*Sum[Sum[n!^2*(Binomial[2*n - 2 - k - j, n - 2] + Binomial[2*n - 2 - k - j, n - 1 - k])/(Max[k, 2]! * Max[j, 2]!), {k, 1, n - 1}], {j, 1, n - 1}], {n, 3, 20}]] (* _Vaclav Kotesovec_, Dec 29 2022 *)
%o (PARI) a(n) = if (n==1, 1, if (n==2, 3, 1 - n^2 + 2*sum(a=1, n-1, sum(b=1, n-1, n!^2*(binomial(2*n-2-a-b, n-2)+binomial(2*n-2-a-b, n-1-a))/(max(a, 2)!*max(b, 2)!))))); \\ corrected by _Michel Marcus_, Dec 29 2022
%K nonn
%O 1,2
%A _Michel Marcus_, Jun 01 2021
%E Offset 1 from _Vaclav Kotesovec_, Dec 29 2022
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